What is the pattern here? Which powers of Phi are also silver means and which silver means are they? [Hint: the answer involves the Lucas numbers.]

What powers of Phi are missing in the answer to the previous question? What are their continued fractions?

Express all the powers of Phi in the form (X+Y√5)/2. Find a formula for Phin in terms of the Lucas and Fibonacci numbers.

Continued Fractions and the Fibonacci Numbers

In this section we will take a closer look at the links between continued fractions and the Fibonacci Numbers.

Squared Fibonacci Number Ratios

What is the period of the continued fractions of the following numbers?

25/9

64/25

169/64

You might have noticed that in all the fractions, both the numerator (top) and denominator (bottom) are square numbers (in the sequence 1, 4, 9, 16, 25, 36 ,49, 64,... ). The numbers that are squared are Fibonacci numbers (starting with 0 and 1 we add the latest two numbers to get the next, giving the series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... ). The fractions above are the squares of the ratio of successive Fibonacci numbers:

25/9 = (5/3)2 = (Fib(5)/Fib(4))2

64/25 = (8/5)2 = (Fib(6)/Fib(5))2

169/64 = (13/8)2 = (Fib(7)/Fib(6))2

...

There is a simple pattern in the continued fractions of all the fractions in this series.

What other continued fraction patterns in fractions formed from Fibonacci numbers (and the Lucas Numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, ... ) can you find?

The surprise in store is what happens if we express this number as a continued fraction. It is

[0; 1, 2, 2, 4, 8, 32, 256,...]

These look like powers of 2 and indeed all of the numbers in this continued fraction are powers of two. So which powers are they? Here is the continued fraction with the powers written in:

[0; 20, 21, 21, 22, 23, 25, 28, ..]

Surprise! The powers of two are the Fibonacci numbers!!!

[0; 2F(0), 2F(1), 2F(2), ... , 2F(i), ...]

Perhaps even more remarkably, a discussion on sci.math newsgroup proves a result that Robert Sawyer posted:- that we can replace the base 2 by any real number bigger than 1 and the result is still true!

A Series and Its Associated Continued Fraction J L Davison, Fibonacci Quarterly vol 63, 1977, pages 29-32. A Simple Proof of a Remarkable Continued Fraction Identity P G Anderson, T C Brown, P J-S Shiue Proceeding American Mathematical Society vol 123 (1995), pgs 2005-2009 has a proof that the Rabbit constant is indeed the continued fraction given above.

References to articles and books

is an early reference to the excellent Rectangle Jigsaw approach to continued fractions that we explored at the top of this page. An even earlier description of this method is found in chapter IV Fibonacci Numbers and Geometry of:

This slim classic is a translation from the Russian Chisla fibonachchi, Gostekhteoretizdat (1951). This classic contains many of the fundamental Fibonacci and Golden section results and proofs as well as a chapter on continued fractions and their properties.

1989, Edward Arnold publishers, ISBN: 0713136618 is an excellent book on continued fractions and lots of other related and interesting things to do with numbers and suggestions for programming exercises and explorations using your computer.

Cambridge University Press, (7th edition) 1999, ISBN: 0521422272 is an enjoyable and readable book about Number Theory which has an excellent chapter on Continued Fractions and proves some of the results we have found above. (More information and you can order it online via the title-link.) Beware though! We have used [a,b,c,d,...]=X/Y as our concise notation for a continued fraction but Davenport uses [a,b,c,d,..] to mean the numerator only, that is, just the X part of the (ordinary) fraction!

Oxford University Press, (6th edition, 2008), ISBN: 0199219869 is a classic but definitely at mathematics undergraduate level. It takes the reader through some of the fundamental results on continued fractions. Earlier editions do not have an Index, but there is a Web page Index to editions 4 and 5 that you may find useful. This latest edition, the 6th, is revised and has some new material on Elliptic functions too.